(i)  Dynamical Stability Surveys
(ii)  Capture Experiments with Growing Planetary Mass
(iii)  Planetesimal-Satellite Interaction during Planetary Migration

Dynamical Stability Surveys (over 1My)

Sun and 4 planets (Jupiter, Saturn,Uranus, Neptune). Initial conditions at the epoch J2000, correction for the barycentrum of the inner solar system. Reference plane is the invariant plane (perpendicular to the total angular momentum of 8 planets -Mercury to Neptune- at epoch J2000)

• 2000 test particles (1000 on prograde and 1000 on retrograde orbits) with initial conditions:

 
• semimajor axis (a) = 0.1, 0.3, 0.5, 0.7, 0.9 * Rh, where Rh is the Hill radius of the planet
• eccentricity (e)       = 0, 0.25, 0.5, 0.75
• inclination (i)            = 0, 20, 40, 60, 80, 100, 120, 140, 160, 180 degrees
• random initial orbital angles, 10 tps with different angles for each a,e,i

 
• MVS (Swift) integrator used with a time step equal to 1/20 of the orbital period of satellites having a = 0.1 Rh (innermost test particles in thesimulation). This was about 0.011, 0.027, 0.077 and 0.15 yr for Jupiter, Saturn, Uranus and Neptune satellites, respectively. The integration time span was 1 My.

 
• Planets were integrated in the heliocentric frame, satellites were integrated in the planetocetric frame.

• for Jupiter: Callisto (R=2403 km, mass~1076 )      is at 0.0125 AU     (mass is in 10^20 g)
• for Saturn: Titan (R=2575 km, mass~1345.5 )       is at 0.00815 AU,    and Iapetus (R=718 km, mass~15.9) is at 0.0237 AU
• for Uranus: Oberon (R=761.4 km, mass~30.14 )   is at 0.00389 AU
• for Neptune: Triton (R=1352.6, mass~214.7)        is at 0.00236 AU

The results are shown as a `bar' code: there are four vertical line segments at each a,i value which lengths are proportional to the number of surviving test particles at four eccentricity values (0, 0.25, 0.5 and 0.75). The full length of 0.1 (in the scale on Y-axis) means that all 10 integrated tps survived. When no particle survives at a,e,i , a dot is plotted at the position of the initial orbit. Observed irregular satellites are also shown in the figures (see the plot of irregular satellites  for more info).

 
These plots show several common characteristics:

1. Stable satellites cannot have inclinations close to 90 degrees. What happens with the tps with i~90 deg is that their orbits have large oscillations due to so-called Kozai cycle.  These are coupled oscillations of inclination and eccentricity with the perihelion argument. The eccentricity oscillations drive the orbit's perihelion to small planetocentric distances where the satellite either impacts the planet or one of the massive satellites. To demonstrate this we calculate a phase portrait for the tps started at a=0.1 Rh, e=0.5, and i =40 degrees in the case of Jupiter. The red and green lines show the eccentricity oscillations induced by Sun calculated analytically in the frame of a simple model, with the Sun on a circular orbit. All satellites with  a=0.1 Rh (of Jupiter) ~0.0355 AU with eccentricities larger than ~0.65 (bold line) are Callisto-crossing (q < 0.0125 AU).  The phase portrait explains why some test particles were elliminated in the numerical survey while others, with the same a,e and i, survived (crosses correspond to the initial orbital elements of elliminated test particles, circles denote the surviving ones). The initial perihelion argument decides the fate.  Either the initial orbital elements fall on the red trajectory and the satellite is condemned to be eliminated because when its perihelion argument equals 90 or 270 deg, it becomes a massive-inner-satellite-crosser, or the initial elements fall on the green trajectory and the eccentricty never gets over the critical limit of  0.65.



  Phase portrait for tps started at a=0.355AU, e=0.5 and i=0
There exists certain asymmetry between prograde and retrograde orbits, the retrograde orbits being generally stable to larger semimajor axes that the prograde ones. It has been shown in thesis of  J. Alvarellos that prograde orbits have larger heliocentric angular momentum than the retrograde ones (at the same planetocentric a). This assigns to the distant prograde satellites (a>0.45-0.48 Rh ) a value of the heliocentric Jacobi constant, which corresponds to `open' zero velocity curves: a prograde satellite with such planetocentric orbit is allowed to escape to a heliocentric orbit.  Conversely, for a distant retrograde satellite (0.5<a<0.74-0.9Rh), the zero velocity curves are `closed', prohibiting escapes (at least in the basic model with the Sun on circular orbit and neglecting all other other planets).  This argument shows that distant prograde satellites MAY escape but do not demonstrate WHY their orbits are chaotic and HOW the escape is achieved.

The following figure shows the orbital elements of one escaping test particle in the numerical survey of Jupiter satellites. This test particle started on prograde orbit with a=0.5 Rh=0.1775 AU, e=i=M=0.


  Orbital elements of a distant prograde satellite affected by evection resonance.

Note the third panel where is the angle computed as the perihelion longitude of the satellite minus the mean longitude of the Sun. This angle librates during most of the seen evolution. In fact, this happens due to so-called `evection' resonance, well known from the theory of the Moon's motion (see also Touma & Wisdom (AJ 115, 1653-1663, 1998). This resonance is responsible for large variations of the eccentricity which eventually drive the satellite outside the planet's Hill sphere. The eccentricty variation and the evolution of the evection angle resemble the way how the secular resonance nu_6 affects asteroidal orbits (Farinella et al. 1994).  If this analogy is as close as we believe, the evection resonance is mathematically decribed by `non-convex' resonant normal form, which permits an indefinite growth of one of the actions. This is why eccentricty grows up to 1 in the simulation. Other prograde test particles started at a=0.5-0.7 Rh behave similarly with shorter escape times at larger semimajor axis. The evection resonance does not occur for retrograde satellites because of their smaller perihelion longitude frequency (Saha and Tremaine 1993).

   Mean motion resonances among Jovian satellites and the Sun

The red lines denote the area occupied by prominent first-order resonances (|kS-k| = 1). Other than first order resonances have negligible (<1e-3AU) widths: their location is shown by almost vertical sequence of points.  The kS=6,k=-1 mean motion resonance which Saha and Tremaine (1993) found to effect the motion of Pasiphae is one of fine resonances  near -0.15 AU (for retrograde orbits: minus the semimajor axis is shown).

This picture is highly idealized description of the real situation because the computation was done in model with Sun on the circular, co-planar orbit. If it were accounted for the eccentricity and inclination of the Sun, each mean motion resonance seen in the picture would `expand' into the multiplet structure of sub-resonances with different combinations of perihelion and nodal arguments. The multiplet structure of a single mean motion resonance would then be localized in a large range of the semimajor axis because, especially for the prograde motion, the perihelion and nodal longitudes are relatively fast angles (unlike the asteroidal case). Anyway, the mean motion resonances does not seem to determine `global' characteristics seen in our `rough' surveys.

Concluding on this point:the effect of the Kozai and evection resonances should result in flat and assymetric distribution of irregular satellites.

Number of surviving test particles vs. time in our rough surveys for different planets:

  Percentage of surviving tps in rough surveys
The color coding: Jupiter (red), Saturn (green), Uranus (blue) and Neptune (white). Two lines for each planet are shown to distinguish between survivals on initially prograde and retrograde orbits. The retrograde orbits -being generally more stable- always correspond to the upper curve. Note that (especially for Jupiter) most test particles escape at t<10^4 yr.

To decide which is the relevant physical model to be use in the surveys, we have performed several additional runs:

• (A) instead of removing a test body when its instantaneous position is within the orbit of Callisto (r<0.0125 AU ), remove it when its perihelion q<R_Jupiter~70 000 km = 0.00046 AU, i.e. when its orbit intesects Jupiter's surface.
• (B) the same as (A ) but with Jupiter's J2 = 14735 10^-6 (Jacobson 2000)
• (C) the same as (A)  but with Ganymede and Callisto as massive bodies (i.e. apart from Jupiter and the Sun, there are 5 massive bodies in simulation: Ganymede, Callisto, Saturn, Uranus, Neptune)

The only region where the statistics of escapes  substantially changes are highly inclined (i>60 deg) orbits with small semimajor axes (a~0.1-0.3 Rh). Eliminating  test particles on Callisto-crossing orbits (standart case) proves to be a reasonable approximation, because both the final statistics (compare with (C)) and the stability/escape boundary are about the same as in the case with Galilean satelites physically present. It is interesting to note that the inclusion of J2 for Jupiter substantially changes the statistics (compare (A) and (B)). J2 decreases the minimum perihelion distance attained during theKozai cycle.

Perspectives:

• run a detailed surveys (escapes and frequency analysis, checking for chaotic behavior and resonances). This survey should answer the question if observed irregular satellites (and the ones that will be discovered in future) sample all available stable trajectories, or choose some particular orbits. In later case, such orbits may be a signature of the mechanism involved in satellites' origin.
• integrate real satellites over 100 My, check on resonant behavior, compute mean orbital elements. The mean elements may serve for identification of satellite families.  It seems that Jupiter retrograde satellites tend

to be more tightly grouped in the mean elements than in the osculating ones. In fact, several sub-groupings of retrograde satellites may be found . A preliminary figure of the retrograde satellites of Jupiter (mean elements = 1000 yr time averages, from Jacobson) look like this:

Osculating and mean elements of Jupiter retrogrades*
 

1) The Carme family has 6 members: J11 Carme (40km diameter), S/2000 J2, J4, J6, J9, J10 (possibly also S/2000_J3)
2) The Ananke family has 3 members: J12 Ananke (30 km diameter) , S/2000 J5, J7
3) Object close to J8 Pasiphae (denoted by `P') is S/2000_J8 .  `S' denotes position of J9 Sinope.

The tight groupings of satellites in the mean elements space strongly suggest their common origin.

The velocity needed to reproduce the full extent of Ananke family in the semi-major axis is 17 m/s, Carme family needs 36 m/s. For comparison, the escape velocities are ~18m/s and ~13.5m/s for Carme and Ananke, respectively (assuming 40km and 30km diameters  and 1.5g/cm3 density). If collisions produced these families, they must have been `gentle' collisions, without much transfer of the linear momentum from the projectile to the target. Alternatively, families may have resulted from gas or tidal disruptions. It might be so, that the above figure also indicates that the spectral heterogenity (Brown, Michael E., AJ 119, 977-983, 2000. Luu, J., AJ 102, 1213-1225, 1991, Sykes et al., Icarus 143, 371-375, 2000) among large retrograde satellites of Jupiter is simply due to the fact that they did not originated from the same body.

*The mean inclinations of Carme, Ananke, Sinope, and Pasiphae shown here are inclinations with respect to the local Laplacian planes, while mean inclinations of other satellites are referred to the ecliptic. I believe, not being sure, that the local Laplacian planes and ecliptic should be almost identical (within ~1 degree, due to the inclination of Jupiter orbit to ecliptics) for distant retrograde satellites, because their secular precessions of node are primarilly driven by the Sun. Thus referring the mean orbital elements of all satellites to the same plane should not make a big difference in the above figure.
 

• continuing with perspectives, one should estimate the probability of collisions.  Any ideas?

 
• it would be interesting to analytically demonstrate  how J2 affects the Kozai cycle and explain the difference between (A) and (B) models.

Capture Experiments with Growing Planetary Mass

1.  Capture by a planet growing in mass. This scenario is closely related to the experiments with temporary gravitational captures, but until now, no one attempted to study it with planetary mass actually growing. The  idea is that the temporary gravitational capture becomes permanent if the planet's mass increase is important during the duration of the temporary capture. This scenario may well apply to Jupiter and Saturn because these planets accreeted large masses within a relatively short period of time, when gas instability developped. The estimated timescale for acreeting an equivalent of 1 Earth mass during this period is 10^2-10^6 years. Uranus and Neptune grew in mass substantially slowlier so that this scenario is more problematic (but still possible) for the origin of their satellites.

 
2.  Capture by gas drag . This scenario proposes that the gas was important for the origin of satellites. There are two possibilities: (i) the capture happened when gas was filling the whole Hill sphere of the protoplanet (before gap openning) or (ii) there was a rather compact disk around the protoplanet at the time of capture (after gap openning). In either case, a stray body (one of many) decelerated in the gas medium just before most of the gas contracted on the protoplanet. Good timing is necessary in this scenario. This could possibly also apply to Uranus and Neptune if putative gas envelope existed around these bodies.

 
3. Capture by collision or swing-by. Two planetesimals on heliocentric orbits encounter or collide within a Hill sphere of the planet and exchange momentum: at least one of them ends up on a planetocentric orbit.

 
4. `Planetary Oort cloud scenario': ejection from the formation site of massive inner satellites. The irregular satellites are remnants of the accretion of the massive inner satellites. They were scattered from by the regular satellites and their perihelia were raised by passing planetesimals. This scenario would presumably preferentially produce prograde orbits, but it fact, no one knows.

The following two figures (make your netscape window large to see them side by side, if possible) show the structure of trajectories (SIGMA_1/1 = mean longitude of a small body minus the mean longitude of Jupiter) around the proto-Jupiter having 30 Earth masses (left) and around Jupiter with its actual mass (right). Six panels per case are shown with different heliocentric eccentricities (0 to 0.16 from (a) to (f)). All shown trajectories lie in the heliocentric orbital plane of Jupiter, which is assumed to have a circular orbit around the Sun. The red curve is where small bodies impacts collide with Jupiter. The domain of retrograde satellites is in the interior of this curve.
 

The set-up of the experiment with growing mass of the Jupiter was the following:

• One planet, Jupiter, exponentially (slower at the beginning, fater in later stages)  increasing its mass from 30 Earth masses to 1 Jupiter mass within 10 000 years (probably faster than what actually happened)

 
• 10 000 test particles on orbits around the Sun, initially uniformly distributed with semimajor axis between  3.8 and 7AU, with 0.01 eccentricity and 0.005 inclination. Random orbital angles.

 
• The Swift-Skeel integrator in heliocentric reference frame for both the planet and test particles with 0.25 yr time step.

 
• Test particle is removed from the simulation if its position or instantaneous Jovicentric perihelion (when being within 1 Hill sphere) becomes smaller than Jupiter radius (~70 000 km = 0.000466 AU).

This experiment resulted in 10 permanent satellites! By permanent satellite we mean a case of a body which orbited Jupiter (had negative planetocentric energy) for at least past 100 yr at t=10 000 y. Visual inspection of the planetocentric orbital elements shows  that all 10 permanently captured bodies were on retrograde orbits! Their orbital elements were the following:
 
 
 

The semimajor axis of Callisto is 0.0125 AU. The satellites which perihelion at t=10 000 yr is smaller than  0.0125 AU (blue color) is, according to our preliminary surveys, unstable.  From the satellites marked in red,  object no. 5210 was Jupiter-grazing at some instant of its orbital history (see last column), which would probably have a destructive effect on its body.  Objects no.  2343 and 6175 are only marginally stable at the end of simulation because of their large eccentricites. The object which most resembles the observed Jovian retrograde satellites is No. 5098. This figure shows its Jovicentric orbital elements after  being captured:
 

  Planetocentric orbital elements of object No.  5098.

This body attains at t=10 000 yr  a slightly smaller semimajor axis than 0.1-0.16 AU which would be typical for a real retrograde satellite, but note that at 7000<t<9000 yr  the semimajor axis was in this range Consequently, if capture occured somewhat  later, the final orbital elements would perfectly mimic the one of a real satellites resembling J12 Ananke.

In total, 30 temporary captures happen with duration > 50 yr. The post-capture planetocentric orbital elements of most of these bodies resemble the ones being captured permanently, but usually the inclination is <130 degrees or the semimajor axis is too large to correspond to a stable orbit. The assymetry of the stability region seen in our surveys seems to cause the preference for captures on retrograde orbits.

I have done a simple calculation: say that from 10 000 bodies between 3.8 and 7 AU only one (No.5210) resembles the orbits of retrograde satellites. The `active' interval of the initial semimajor axis where long-term captures occur is 4.5-6 AU (see Table ). Say that there is 1 good capture from 4700 test particles initially started at 2.3-6 AU. The total mass of solids in ring of 1.5 AU width at a=5 AU could have been something like 2-3.5 Earth masses. If the diferential size distribution was something like D^-3, then there were of order of 10^8-10^10 bodies with D > 4km (all 14 known retrograde satellites of Jupiter have D>4km). If we assume, according to our experience with the satellite capture due to mass growth of the planet on 10^4 years, a probability of capture  ~ 10^-4  - 10^-5, we create from 1000  to 10^6 retrograde satellites. Apparently, this is too much and probably indicates that our model not good. Slower growth of the planet's mass (10^5-10^6 yr) or the presence of the gas probably decreases the capture probability.  On the other hand, as Luke pointed to me, observed satelites seem to have rather shallow size-distribution (~D^-2)., so that considering  D^-3 was probably not correct.One should attempt to make more serious estimates.

We conclude that our simple model with fast-growing mass of Jupiter seems to be too efficient! Good :-)
 

Perspectives:

1. Run several experiments varying the timescale of the planets growth. Is there any chance to apply this to Uranus or Neptune?  Possibly also change initial eccentricties and inclinations.

 
2. Account for the gas friction, with or not varying planet mass. Consider both possibilities: (i) that full Hill sphere of the planet is filled by the gas or (ii) that exists a compact gas disk near the planet.

 
3. We need to resolve the problem with the integrator speed when increasing the number of particles. For slower planetary mass growth, I would like to simulate 10^5 test particles.

 

Planetesimal-Satellite Interaction During Planetary Migration

This project has been recently started with F.  Roig (visiting SwRi) and C. Beauge (staff scientist of INPE laboratory, San Jose dos Campos, Brazil). The idea is the following:

During the phase of the planetary migration, planets interacted with a large number of massive planetesimals. For example, Neptune is though to migrate as much as 7 AU. When a massive planetesimal penetrates into the Hill sphere of a planet, it certainly disturbes existing satellites. The question is how many massive planetesimals are needed to significantly change the orbit of the satellites and possibly detsabilize them. Does this set some constraint on the migration? It is hard to imagine that Neptune could have migrated by 7 AU without having lost some of its satellites due to the planetesimals gravitational sweeping. Satellites of other planets might have been also affected.

In the preliminary phase, we divide the problem in two stages:

(1) We first take the Sun and 4 outer planets on their present orbits (massive bodies) and 1000-10000 planetesimals (massles particles) initially distributed in a cold disk. We advance this system for  10-50 My and create a database of close encounters: each time a planetesimal is within a Hill sphere, its positions and velocities are registered.  Later on, we assume masses for planetesimais and by the principle of action and reaction compute, from the energy and momentum balance of planetesimal's orbit, the change of planetary semimajor axis. The simulation of Hahn and Malhotra will be used as a reference. One of the nice features of this schema is that only one integration is done and different disk masses and density distribution can be considered aposteriori.

(2) In planetocentric reference frame, advance 100-1000 satellites at different semimajor axis on their Keplerian orbits, until the first planetesimal enters within a Hill sphere. Interpolate the trajectory of the planetesimal between the points registered in the database of phase I and simulate its effect on the (massless) satellites. After planetesimal exists, advance satellites on Keplerian orbits towards the next encounter. This should be fast. The expected effect on satellites is a slow random walk of their orbital elements with possibly more violent evolution during very close planetesimal encounters.
 

Perspectives:

1. If the preliminary phase turns out to be interesting and we gain some feeling about the parameters in game, we'd like to perform a more realistic simulation, possibly using Symba.

 
2. The database obtained in phase I may serve for other studies. A particularly interesting one is to estimate the number of collisions of two planetesimals within a planetary Hill sphere to gain some insight on the efficiency of this mechanism for satellite formation. It is also related to the Oort cloud scenario of the origin of irregulars. We can simulate a large number of bodies being scattered by massive inner satellites and include (from the database of phase I)  the passing planetesimals. This will give us an insight on the efficiency of passing planetesimals to lift the perihelion distances.